I found a thread on GW that has a picture of a white octagonal mosaic with burgundy accents, scroll down until you see the picture. I think you are going to need to incorporate some of the elements already in the bathroom, and the tile trim color seems the most logical to carry through to the floor either with an accent in burgundy, or if you could find a burgundy floor tile- tile the entire floor with it. Here is a link that might be useful: octagonal tile with burgundy trim

I adore arabesque (and lantern. and other spanish -influenced traditional shapes-after all it's my favorite style ever, if I have to choose) but it doesn't spell modern to me.depends of course of what one means by modern because often people mean different things by that -some mean certain style, some mean something being more recent, contemporary. here's Fireclay tile blog..I liked glazed bricks on the first pic..second as well-shapes but less fluid shapes, more geometrical ones..and I strangely like this sweater pattern..I think last from the top? the one they call prep school. it might be too much and one needs to make sure they have enough space on their backsplash to pull it off. yet it exudes certain warmth..as if it was a sweater..:) https://www.fireclaytile.com/blog/ (as for the brand itself. I never used Fireclay myself, it's not inexpensive either, yet many people on these forums did, and remained extremely pleased with their quality. take a look at what they offer..they have many interesting products. including a lot of black plus white designs, some with a lot of contrast in the pattern, some, paler, more "barely there" of an effect)

New light paint color for the walls pulled from the tile will make a big difference. I would price replacing the counter with a lighter, quieter prefab laminate counter. They aren't terribly expensive and might be within your budget. A six-foot version is around $100 plus install. The color choices are limited. Greys and browns are tricky so be sure to bring a sample home to look at in your light to make sure the undertones work together. https://www.homedepot.com/p/Hampton-Bay-8-ft-Laminate-Countertop-in-Sea-Salt-with-Tempo-Edge-and-Integrated-Backsplash-011312010819529/305470953

I think that's a good idea. you can de-gloss them (they have a deglossing product at home depot or amazon) and then give them a decent sanding with 150/220 grit paper. clean off w/mineral spirits. get a spray primer in the can like Zinsser oil base primer a give a full, but light coat. when dry, fine sand easily w/220or 320 grit. wipe clean. Now you can paint. Use BM Advance paint. I've painted many wood furniture pieces and this paint has the best finish. As for tile, give me a hint of what design you like and i'll give you some picks. Also, can you post a pic of the whole room (the vanity wall and any other walls?) another budget idea is to check w/stone yards or the fabricators warehouse and ask him if he has any remnant stone tops. a lot of time they will trash those smaller pieces. you need something in 50"X24" range. (it's a 48" vanity, yes? ) Also, check craigslist. you can find some good deals on leftovers. and check stone yard or larger tile place for pre-fab countertops. my place has a ton of these quartz, marble granite,,, they're already finished and ready to fit on top of vanities want something a bit more dramatic and bold? Pick the black quartz w/the white veining, do a black and white floor tile bring in some black metal faucet and sconces, and do a white tile in back of the vanity if there's a shower in there, match the tile in the shower w/the quartz top paint the vanity black, do a white quartz/marble top or paint your vanity like this. basketweave marble tile want bold? pick a bold color for the vanity. and do a full on wall of these matte black subway tiles. (if you could stain your vanity slightly darker and change out your hardware, you could get this look) another one w/similar wood tone. do an entire wall of tile, pick a cool floor tile, sconces